If they doctor failed to act as a reasonable doctor in the community would have under the circumstances, thus causing damages, then there is malpractice.
In this case, what are your actual damages? Were you asleep during surgery. Did you suffer additional pain, suffering, loss of enjoyment of life, emotional distress, and out of pocket expenses?
Your question is really two in one.
Who is "responsible" for the fees? If by who has to pay, the answer is you. You contracted with the doctor and the anesthesiologist for their services. The anesthesiologist did his/her job by keeping you under while the procedure was performed. They're entitled to be paid. The contract you signed with the hospital contained language indicating you were responsible for the entire bill, I'm sure.
Wrapped up in there, I assume, is the question of can you sue the doctor that performed the cosmetic procedure for the money you have to pay - at least the money above and beyond what the original cost was?
My short answer is.... yes, but.
Yes, you can file suit for just about anything you want, BUT... If you think there were "damages" as a result of medical malpractice by your doctor, you can sue. Ordinarily, damages are for resulting harm, failure to properly diagnose and/or treat or other "bad" outcomes as the result of malpractice by a physician. In your situation, you're asking the doctor to pitch in and pay for the "extra" charges you incurred as a result of malpractice.
Here's the big issue, though: Proving malpractice. In every surgical procedure, there are always risks. I assume whatever procedure was performed carries with it a certain amount of risk - as does any surgical procedure. Stitches can fail without malpractice. Surgery can take longer than thought initially, requiring extra time on the table.
In order to prevail, you will have to show that the doctor didn't perform the procedure following the standard of care within the medical community. You would need to find another doctor qualified in this field to review the records and certify the lawsuit, verifying that there was malpractice. Then becomes the battle of the experts. Your doctor will have a handful of doctors to testify on his behalf that his actions were proper and this was an unexpected, but not negligent, complication during the procedure and the doctor handled things properly. You'll have your experts to say the opposite - that the doctor was negligent, leading to the "extra" time on the table.
You'll spend thousands and thousands and thousands of dollars pursuing this lawsuit (if anyone would take it and you could get it certified) for what damages? How much was the "additional hospital and anesthesia time fees?"
If there was a bad result from the surgery (it worsened the condition), my answer would be completely different. From what you've described here, I just don't see a lawsuit. If you feel strongly about this, consult with a couple of personal injury or medical malpractice attorneys in your area and run it by them.
*** I do not practice PI or medical malpractice, so my answer is based on general principles of law. It is meant for general information only. This is not intended to be a legal opinion, nor does it create an attorney-client relationship between us. You may have a perfectly valid lawsuit and act according to your best interests to protect your legal rights. Some lawsuits have specific time requirements that must be adhered to or you will lose the ability to file suit. Take my answer with a grain of salt and contact a qualified attorney in your area to protect your legal interests.
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